Generation of Fermat’s spiral patterns by solutal Marangoni-driven coiling in an aqueous two-phase system

The solutal Marangoni effect is attracting increasing interest because of its fundamental role in many isothermal directional transport processes in fluids, including the Marangoni-driven spreading on liquid surfaces or Marangoni convection within a liquid. Here we report a type of continuous Marangoni transport process resulting from Marangoni-driven spreading and Marangoni convection in an aqueous two-phase system. The interaction between a salt (CaCl2) and an anionic surfactant (sodium dodecylbenzenesulfonate) generates surface tension gradients, which drive the transport process. This Marangoni transport consists of the upward transfer of a filament from a droplet located at the bottom of a bulk solution, coiling of the filament near the surface, and formation of Fermat’s spiral patterns on the surface. The bottom-up coiling of the filament, driven by Marangoni convection, may inspire automatic fiber fabrication.


Supplementary Calculations Force provided by Marangoni convection.
Based on the motion of the red PS particles (diameter 22 µm, density 1.050 g cm −3 , volume 5.6×10 −15 m 3 , weight 5.88×10 −9 g) in Supplementary Movie 5, we estimated the force provided by Marangoni flow. According to Newton's second law: (2) Where F is the sum of forces applied to the particle, including the force provided by Marangoni flow ( M), buoyancy ( ), gravity (G) and viscous force ( v) (Supplementary Fig. 14a). m is the mass of the particle and a is the acceleration at y direction, and the maximum value of a is 0.18 mm/s 2 at t = 107.1 s (Supplementary Fig. 14b). So, the maximum value of ma is 1.05×10 −6 nN. The gravity (G) can be calculated by G = m g = ρ 4/3 πR 3 g

(3)
Where ρ is the density of the PS particle (1.050 g cm −3 ), R is the radius of the particle (11 µm) and g is gravitational acceleration (9.8 m s −2 ). So, the Gravity is 5.74×10 −2 nN. The buoyant force can be calculated by Fb = ρl V g = ρl 4/3 πR 3 g (4) where ρl is the density of PEG-SDBS solution (1.010 g mL −1 ). So, the buoyant force is 5.52×10 −2 nN. The viscous force can be estimated by the Stokes equation 1

Fv = 6πRηU
(5) where η is the viscosity of PEG-SDBS solution surrounding the droplet (~ 10 −2 Pa s), U is the velocity of this particle at y direction. As is shown in Supplementary Fig. 14c, U is changing with time and its maximum value is 0.35 mm/s at t = 107.8 s. So, the maximum viscous force for this particle is estimated to be 0.7 nN. Since Fvmax is more than two orders of magnitude larger than ma and G − Fb, Mmax ≈ Fvmax (6) So for the 22-µm PS particles in the flow field at t = 107.8 s, the Marangoni driving force is 0.7 nN. Similarly, for the particles/droplets (diameter ~ 1 µm) in the filament, the maximum Marangoni driving force is estimated to be 0.03 nN, according to Equations (5) and (6). Then we obtain a series of Umax and FMmax by analyzing the upward motion of particles at different times. We find that both Umax and FMmax decrease linearly with time, consistent with the linear decrease of angular velocity of spiral patterns with time on the surface of the PEG-SDBS solution. We also notice that, the position of the Umax point is also changing with time, which is probably resulted from the oscillation of the filament. Similarly, we also analyze the Marangoni force in the system with a bulk solution depth of 3.4 mm. We find that particles in this system have similar Umax(t) and FMmax(t) but larger values, compared to those obtained in the system with a depth of 2.5 mm ( Supplementary Fig. 14d-f).

Model problem for CaCl2 concentration around a moving filament.
As the filament moves horizontally, the CaCl2 it contains diffuses outward into the ambient fluid and is advected by the flow in that fluid. Let the horizontal velocity of the filament be U, and let the radius of the filament be a. The diffusivity of CaCl2 in water is D. At the beginning of the experiment, the Peclet number Pe = U a/D ≫ 1, which means that gradients of CaCl2 are confined to a thin boundary layer of thickness δ ≪ a around the filament.
For simplicity, we ignore the radius of curvature of the spiral arms and consider the filament to be an infinitely long cylinder. Supplementary Fig. 15 shows a cross-section of the filament. The concentration c(r, θ) of CaCl2 in the boundary layer satisfies the boundary-layer equation ( + ) = (7) where u and v are the tangential and radial components of the velocity, respectively. In (7) and henceforth, velocities are measured in units of U and distances in units of a. Eqn. (7) can be solved analytically for the cylindrical geometry at hand. We first apply the Von Mises transformation, whereby the radial coordinate is replaced by the stream function ψ. The result is = . (8) Now we turn to the solution for the two-dimensional flow around an infinite cylinder at low Reynolds number, which is given on pp. 244-245 of Reference 2. 2 Expanding this solution in powers of the distance ζ = r − 1 away from the cylinder's surface, we find = sin , = −1/2 sin , At this point, we assume that the concentration on the surface of the cylinder is uniform and equal to αc0, where c0 is the CaCl2 concentration in the interior of the filament. Equation (14) then admits a similarity solution c = c(η) in terms of the similarity variable η = −ψ/t 2/3 . Substituting this form into (14), we obtain where primes denote derivatives with respect to η. Finally, we set = , whereupon (15) becomes The solution of (16) is where A and B are determined by the boundary conditions c(0) = αc0 and c(∞) = 0. We find Finally, the variable z is related to the original variables ζ and θ by Now we are only interested in the vertical concentration profile at θ = π/2, where G(π/2) ~ 1.198. Moreover, because z as given by (19) varies only slightly as a function of Re, we choose C = 0.554 for a typical experimental Reynolds number Re = 0.1. We then have (20) Supplementary Fig. 16 shows vertical profiles of CaCl2 concentration above the filament for two Peclet numbers Pe = 60 and 30. We chose α = 0.5 because diffusion of CaCl2 occurs also within the filament, so that the concentration on the boundary is intermediate between zero and c0. In constructing Supplementary Fig. 16, we first fixed c = 0.001c0 at the surface for the Pe = 60 case ( Supplementary Fig. 16a). As discussed in the main text, 0.001c0 is the order of magnitude of the concentration difference that drives the very small flow velocities (≈ 0.5 mm s -1 ) measured in the experiments. Because Pe = 60 greatly exceeds unity, concentration gradients are mostly confined to a boundary layer of thickness δ ≪ a. The concentration therefore decreases rapidly with height, diminishing to 0.001c0 on the surface at a height 0.985a above the filament. Next, we turned to the Pe = 30 case and sought the depth of the filament required to yield a smaller surface concentration c = 0.0005c0. That depth is 1.287a, 30% greater than for Pe = 60. Supplementary Fig. 16 implies a positive feedback loop in which the dense filament slowly sinks, thereby reducing the surface concentration of CaCl2 and the associated Marangoni force, which in turn slows down the flow (decreasing Pe) and causes the filament to sink even more. However, this positive feedback loop in principle works equally in the other direction, with a rising filament increasing the surface concentration of CaCl2 and increasing the flow speed, causing the filament to rise even more. Why then is this second alternative not observed? The answer is diffusion, which tends to equalize the surface concentration of CaCl2 and reduce the driving Marangoni force. Movies of the experiments show clearly that the outer arms of the spiral become progressively more diffuse with time, which corresponds to a reduction in the surface gradients of CaCl2 and a progressive slowing down of the flow. Diffusion is therefore the ultimate reason why the experimental flow is observed to slow down rather than to speed up.

Supplementary Tables
Supplementary Table 1